منتدى استراحات زايد - حل واجب m132 : 0544321455 : 00966544321455 : a_al

M132: LINEAR ALGEBRA
Q−1: [5×2 marks]
Answer each of the following as True or False (justify your answer):

a) Two matrices and are row equivalent.
b) The linear system is inconsistent.
c) If A is a nonsingular matrix such that A4 = AT, then |A| = 1.
d) If A and B are nonsingular symmetric matrices, then ABA-1 is symmetric.
e) The vector is a linear combination of and .
Q−2: [2+2+1 marks] Let . Compute, if possible, a) (ABT)T, b) A − 2BTB, c) (ATB)T.
Q−3: [5 marks] Find all values of a for which the following linear system:
a) has no solution, b) has a unique solution, c) has infinitely many solutions.
Solve the linear system for a = 4.
Q¬−4: [3+1+1 marks] Let , and .
a) Find A-1.
b) Find a matrix X such that AX + B = C.
c) Is it possible to find a matrix Y such that YA + B = C? Explain your answer.
Q¬−5: [5 marks] Let and assume that |A| = 10. Find the determinant of .
Q−6: [5 marks] Find all values of a for which is linearly dependent.
Q¬−7: [5 marks] Let X1, X2 and X3 be three linearly independent vectors in Rn and let Y1 = X1 + X2, Y2 = 2X2 − X3 and Y3 = X1 + X2 − 2 X3. Show that Y1, Y2 and Y3 are linearly independent vectors in Rn.